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Why not both true and false?Rather than merely question the validity of mathematics, I suggest that you should conduct an exhaustive investigation of this. If you can prove that this is correct your name will be famous in scientific and mathematical circles. And if you're wrong your situation will be no different than it is today. Go to, sir! Bon chance!

Notation for powers of trigonometric functions+1 for "it's just a convention". You can use whatever notation you care to use for whatever you care to use it for - you'll only have problems if you try to communicate with someone else. The Nobel Prize-winning physicist Richard Feynman developed his own notation for trigonometric functions because, as a self-taught mathematician, he did not initially know the standard notation. Sadly, I am not Richard Feynman, I am not a self-taught mathematicians who conjured trigonometry from thin air while in high school, and thus it's a good idea for me, at least, to use the standard notation for trig.

Mathematicians ahead of their time?I suspect that if any one individual had not lived, someone else would have made the same discoveries. Perhaps not all the discoveries made by X would then have been made by Y if X had not lived; perhaps it would have been Y, Z, A', and Q'' that would have made those discoveries. In mathematics the truth is, so to speak, out there - it's a matter of turning over the rocks until something bites you. :-) Share and enjoy.

Can a piece of A4 paper be folded so that it's thick enough to reach the moon?@CompuChip - you and I made the same mistake. Folding the paper 42 times doesn't give 42 layers (or 42 sheet thicknesses) - it gives something like 4.4x10^12th layers (2 to the 42nd). That's a whole lotta layers - enough so that if a piece of paper is about 1/10th of a millimeter thick then all those layers would add up to about the average orbital radius of the moon (384000 km, give or take). However, in practice a piece of paper can't be folded that many times, which is good job-wise if you happen to be a rocket scientist. :-) Share and enjoy.

Monty hall problem extended.@mousomer: one of our funniest experiences was transporting a goat kid home in the back of a Subaru. Driving up I-77 we got a lot of really odd looks from people. (As one car passed us the woman in the passenger seat looked at us and I clearly saw her say, "What the f...?" :-). Don't think I ever laughed that much while driving, before or since - an' I swear, I was sober as a judge..! :-)

Dec17

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Monty hall problem extended.Goats are browsers; they'd prefer it if your yard was full of trees, bushes, brambles, and stuff they can reach up to. Although they'll graze if that's what's available they do better and stay healthier on browse.